[1]}, [2]} studied
penalized nonparametric regression methods where the penalty can be
seen as a variant of the right-hand side of (REF ) with the
\(L_1\)  norms replaced by the squared \(L_2\)  norms. Also, the anchoring at
\(\mathbf {0}\)  in our treatment is handled differently in
[1]}, [2]} by means of averaging. The replacement of the \(L_1\) 
norms by the squared \(L_2\)  norms reduces the class
of considered functions significantly, and in particular, the methods
of [1]}, [2]} will not work
for fitting functions in \(\mathcal {F}_{\text{mars}}^{d}\) . [7]}
studied penalized nonparametric regression for a class of
penalties characterized in terms of certain hierarchical notions of total
variation. Their hierarchical total variations are defined using a
pre-fixed grid of points, and for functions \(f\)  that are sufficiently
smooth, one of their hierarchical total variations (corresponding to
\(m = 2\)  in their notation) converges to
\(V_{\mathrm {mars}}(f)\)  as the grid resolution becomes arbitrarily
small. [8]} studied multivariate generalizations
of trend filtering in the discrete setting where the goal is to
estimate a multivariate tensor. Their method is related to the discrete
formulation of our estimator described in Proposition
REF  (see Remark
REF  for more details). [9]}
discussed function classes similar to \(\mathcal {F}_{\infty -\text{mars}}^d\)  and norms similar
to \(V_{\mathrm {mars}}(\cdot )\)  and used them for estimation in settings that seem
different from our classical nonparametric regression framework.
